Compute Distributional Statistics from a Discrete Random Variable

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FFT_DISCT_RAND_VAR_STATS compute mean, sd, percentiles
Default
Parse Parameters
f(y), f(c), f(a): Compute Scalar Statistics for outcomes
f(y), f(c), f(a): Compute Distributional Statistics for outcomes
Collect Statistics
Display

function [ds_stats_map] = fft_disc_rand_var_stats(varargin)

FFT_DISCT_RAND_VAR_STATS compute mean, sd, percentiles

Model simulation generates discrete random variables, to analyze, need to calculate various statistics

Statistics include:

$\mu_Y = E(Y) = \sum_{y} p(Y=y) \cdot y$
$\sigma_Y = \sqrt{ \sum_{y} p(Y=y) \cdot \left( y - \mu_y \right)^2}$
$p(y=\min(y))$
$p(y=\max(y))$
percentiles: $min_{y} \left\{ P(Y \le y) - percentile \mid P(Y \le y) \ge percentile \right\}$
fraction of outcome held by up to percentiles:

@param st_var_name string name of the variable (choice/outcome) been analyzed

@param ar_choice_unique_sorted array 1 by N elements in the sample space of the discrete random variable ordered. Unique consumption values ordered. Unique asset choices ordered. etc.

@param ar_choice_prob array 1 by N probability mass function associated with each element ar_choice_unique_sorted

@param ar_fl_percentiles array 1 by M some vector of percentiles (0 to 100) that we would like to compute based on the discrete random variable's probability mass function and x values.

@return ds_stats_map container various distributional statistics

@example

% run function
st_var_name = 'binom';
[ds_stats_map] = fft_disc_rand_var_stats(st_var_name, ar_choice_unique_sorted, ar_choice_prob, ar_fl_percentiles);
% retrieve scalar statistics
% fl_choice_mean = ds_stats_map('fl_choice_mean');
fl_choice_sd = ds_stats_map('fl_choice_sd');
fl_choice_prob_zero = ds_stats_map('fl_choice_prob_zero');
fl_choice_prob_max = ds_stats_map('fl_choice_prob_max');
% retrieve distributional array stats
ar_choice_percentiles = ds_stats_map('ar_choice_percentiles');
ar_choice_perc_fracheld = ds_stats_map('ar_choice_perc_fracheld');

Default

use binomial as test case

fl_binom_n = 30;
fl_binom_p = 0.3;
ar_binom_x = 0:1:fl_binom_n;

% f(x)
ar_choice_prob = binopdf(ar_binom_x, fl_binom_n, fl_binom_p);
% x
ar_choice_unique_sorted = ar_binom_x - 10;
% percentiles of interest
ar_fl_percentiles = [0.1 1 5:5:25 35:15:65 75:5:95 99 99.9];

% display
st_var_name = 'binom';

% display
bl_display_drvstats = true;

% default
default_params = {st_var_name ar_choice_unique_sorted ar_choice_prob bl_display_drvstats ar_fl_percentiles};

Parse Parameters

[default_params{1:length(varargin)}] = varargin{:};
[st_var_name, ar_choice_unique_sorted, ar_choice_prob, bl_display_drvstats, ar_fl_percentiles] = default_params{:};

f(y), f(c), f(a): Compute Scalar Statistics for outcomes

Compute these outcomes:

mean: $\mu_y = \sum_{y} p(Y=y) \cdot y$
sd: $\sigma_y = \sqrt{ \sum_{y} p(Y=y) \cdot \left( y - \mu_y \right)^2}$
prob(outcome=0):
prob(outcome=max(outcome)): $p(y=\max(y))$

% Mean of discrete random variable
fl_choice_mean = ar_choice_prob*ar_choice_unique_sorted';
% SD of discrete random variable
fl_choice_sd = sqrt(ar_choice_prob*((ar_choice_unique_sorted'-fl_choice_mean).^2));
% Coef of Variation of discrete random variable
fl_choice_coefofvar = fl_choice_sd/fl_choice_mean;
% min of y from policy function, p(y) might be 0
fl_choice_min = min(ar_choice_unique_sorted);
% max of y from policy function, p(y) might be 0
fl_choice_max = max(ar_choice_unique_sorted);
% prob(outcome=min(outcome)), fraction of people not saving for example
fl_choice_prob_min = sum(ar_choice_prob(ar_choice_unique_sorted == min(ar_choice_unique_sorted)));
% prob(outcome=0), fraction of people not saving for example
fl_choice_prob_zero = sum(ar_choice_prob(ar_choice_unique_sorted == 0));
% prob(outcome<0), fraction of people borrowing
fl_choice_prob_below_zero = sum(ar_choice_prob(ar_choice_unique_sorted < 0));
% prob(outcome>0), fraction of people borrowing
fl_choice_prob_above_zero = sum(ar_choice_prob(ar_choice_unique_sorted > 0));
% prob(outcome=max(outcome)), fraction of people saving up to max of grid,
% in principle if this is large, need to increase grid max value
fl_choice_prob_max = sum(ar_choice_prob(ar_choice_unique_sorted == max(ar_choice_unique_sorted)));

f(y), f(c), f(a): Compute Distributional Statistics for outcomes